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  "body": "\n## Summary\n\nLittlewood–Paley theory splits a function on $\\R^n$ into pieces whose [[Fourier transforms|DPVM34ADZV]] live in dyadic annuli $\\set{2^{j-1} \u003c \\abs{\\xi} \u003c 2^{j+1}}$, by the [[Littlewood–Paley decomposition|D1PAT0K446]].\nIts main result is the [[Littlewood–Paley inequality|6GYQNB2862]].\nIt says that, for $1 \u003c p \u003c \\infty$, the $L^p$ norm of $f$ is comparable to that of its [[square function|H4E1PW67Z1]] $\\left( \\sum_j \\abs{\\Delta_j f}^2 \\right)^{1/2}$.\nSo the dyadic pieces behave in $L^p$ almost like orthogonal pieces in $L^2$.\nThe theory underlies multiplier theorems, the description of function spaces such as Sobolev, Besov and Triebel–Lizorkin spaces, and paraproduct decompositions of products.\n\n## Explanations\n\n### L1 human\n\nThink of a sound signal split into octaves: low notes, then notes one octave higher, then the next octave, and so on.\nIn $L^2$, the energy of the signal is the sum of the energies of the octaves, by Plancherel's theorem.\nLittlewood–Paley theory says that a similar statement survives for the $L^p$ norms with $1 \u003c p \u003c \\infty$.\nOne only has to combine the octaves through a square root of a sum of squares.\nFor example, changing the signs of the octave pieces in any way only changes the $L^p$ norm by a bounded factor.\n",
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